CBSE Class 12 Mathematics Vectors Algebra MCQs Set 09

Question. If \(\vec{a}\) and \(\vec{b}\) are two unit vectors inclined to x-axis at angles \(30^\circ\) and \(120^\circ\) respectively, then \(|\vec{a} + \vec{b}|\) equals
(a) \(\frac{2}{\sqrt{3}}\)
(b) \(\sqrt{2}\)
(c) \(\sqrt{3}\)
(d) 2
Answer: (b) \(\sqrt{2}\)

Question. If \(\vec{b}\) and \(\vec{c}\) are any two non-collinear unit vectors and \(\vec{a}\) is any vector, then find the value of \((\vec{a} \cdot \vec{b})\vec{b} + (\vec{a} \cdot \vec{c})\vec{c} + \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{|\vec{b} \times \vec{c}|^2} (\vec{b} \times \vec{c})\).
(a) \(\vec{a} + \vec{b} + \vec{c}\)
(b) \(\vec{c}\)
(c) \(\vec{a}\)
(d) \(\vec{b}\)
Answer: (c) \(\vec{a}\)

Question. If \(\vec{a}\) and \(\vec{b}\) are unit vectors enclosing an angle \(\theta\) and \(|\vec{a} + \vec{b}| < 1\), then
(a) \(\theta = \frac{\pi}{2}\)
(b) \(\theta < \frac{\pi}{3}\)
(c) \(\pi \ge \theta > \frac{2\pi}{3}\)
(d) \(\frac{\pi}{3} < \theta < \frac{2\pi}{3}\)
Answer: (c) \(\pi \ge \theta > \frac{2\pi}{3}\)

Question. If \(|\vec{a}| = 10\), \(|\vec{b}| = 2\) and \(\vec{a} \cdot \vec{b} = 12\), then the value of \(|\vec{a} \times \vec{b}|\) is
(a) 5
(b) 10
(c) 14
(d) 16
Answer: (d) 16

Question. If \(\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}\) and \(\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}\), then \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\) are
(a) parallel
(b) perpendicular
(c) skew
(d) None of the options
Answer: (b) perpendicular

Question. Area of a parallelogram whose adjacent sides are represented by the vectors \(2\hat{i} - 3\hat{k}\) and \(4\hat{j} + 2\hat{k}\) is
(a) \(4\sqrt{14}\) sq. units
(b) \(2\sqrt{7}\) sq. units
(c) \(4\sqrt{7}\) sq. units
(d) \(4\sqrt{19}\) sq. units
Answer: (a) \(4\sqrt{14}\) sq. units

Question. The direction ratios of the vector \(3\vec{a} + 2\vec{b}\), where \(\vec{a} = \hat{i} + \hat{j} - 2\hat{k}\) and \(\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}\) are
(a) 7, 5, 4
(b) 7, -5, 4
(c) -7, 5, 4
(d) 7, 5, -4
Answer: (b) 7, -5, 4

Question. The angle between two vectors \(\vec{a}\) and \(\vec{b}\) with magnitudes \(\sqrt{3}\) and 4, respectively and \(\vec{a} \cdot \vec{b} = 2\sqrt{3}\) is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{3}\)
(c) \(\frac{\pi}{2}\)
(d) \(\frac{5\pi}{2}\)
Answer: (b) \(\frac{\pi}{3}\)

Question. The position vector of the point which divides the joining of points \(2\vec{a} - 3\vec{b}\) and \(\vec{a} + \vec{b}\) in the ratio 3 : 1 is
(a) \(\frac{3\vec{a} - 2\vec{b}}{2}\)
(b) \(\frac{7\vec{a} - 8\vec{b}}{4}\)
(c) \(\frac{3\vec{a}}{4}\)
(d) \(\frac{5\vec{a}}{4}\)
Answer: (d) \(\frac{5\vec{a}}{4}\)

Question. If \(|\vec{a}| = 4\) and \(-3 \le \lambda \le 3\), then the range of \(|\lambda\vec{a}|\) is
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]
Answer: (c) [0, 12]

Question. If \((2\hat{i} + 6\hat{j} + 27\hat{k}) \times (\hat{i} + p\hat{j} + q\hat{k}) = \vec{0}\), then the values of p and q are
(a) \(p = 6, q = 27\)
(b) \(p = 3, q = \frac{27}{2}\)
(c) \(p = 6, q = \frac{27}{2}\)
(d) \(p = 3, q = 27\)
Answer: (b) \(p = 3, q = \frac{27}{2}\)

Question. If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors such that \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), then write the value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\).
(a) \(\frac{3}{2}\)
(b) 3
(c) \(-\frac{3}{2}\)
(d) \(-3\)
Answer: (c) \(-\frac{3}{2}\)

Question. Find the value of \(\lambda\) for which the vectors \(3\hat{i} - 6\hat{j} + \hat{k}\) and \(2\hat{i} - 4\hat{j} + \lambda\hat{k}\) are parallel.
(a) \(\frac{2}{3}\)
(b) \(-\frac{3}{2}\)
(c) \(-\frac{2}{3}\)
(d) \(\frac{3}{2}\)
Answer: (a) \(\frac{2}{3}\)

Question. Find the value of \(\lambda\) so that the vectors \(2\hat{i} - 4\hat{j} + \hat{k}\) and \(4\hat{i} - 8\hat{j} + \lambda\hat{k}\) are perpendicular.
(a) 20
(b) \(-40\)
(c) 40
(d) \(-20\)
Answer: (b) \(-40\)

Question. Find the projection of the vector \(\hat{i} + 3\hat{j} + 7\hat{k}\) on the vector \(2\hat{i} - 3\hat{j} + 6\hat{k}\).
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (a) 5

Question. The vector having initial and terminal points as \((2, 5, 0)\) and \((-3, 7, 4)\) respectively is
(a) \(-\hat{i} + 12\hat{j} + 4\hat{k}\)
(b) \(5\hat{i} + 2\hat{j} - 4\hat{k}\)
(c) \(-5\hat{i} + 2\hat{j} + 4\hat{k}\)
(d) \(\hat{i} + \hat{j} + \hat{k}\)
Answer: (c) \(-5\hat{i} + 2\hat{j} + 4\hat{k}\)

Question. The vectors from origin to the points A and B are \(\vec{a} = 2\hat{i} - 3\hat{j} + 2\hat{k}\) and \(\vec{b} = 2\hat{i} + 3\hat{j} + \hat{k}\), respectively, then the area of triangle OAB (in sq. units) is
(a) \(\sqrt{340}\)
(b) \(\sqrt{325}\)
(c) \(\sqrt{229}\)
(d) \(\frac{1}{2}\sqrt{229}\)
Answer: (d) \(\frac{1}{2}\sqrt{229}\)

Question. If \(|\vec{a}| = 3, |\vec{b}| = 4\), then the value of \(\lambda\) for which \(\vec{a} + \lambda\vec{b}\) is perpendicular to \(\vec{a} - \lambda\vec{b}\), is
(a) \(\frac{9}{16}\)
(b) \(\frac{3}{4}\)
(c) \(\frac{3}{2}\)
(d) \(\frac{4}{3}\)
Answer: (b) \(\frac{3}{4}\)

Case Based MCQs

Case I : Read the following passage and answer the questions.
A barge is pulled into harbour by two tug boats as shown in the figure.

Question. Position vector of A is
(a) \(4\hat{i} + 2\hat{j}\)
(b) \(4\hat{i} + 10\hat{j}\)
(c) \(4\hat{i} - 10\hat{j}\)
(d) \(4\hat{i} - 2\hat{j}\)
Answer: (b) \(4\hat{i} + 10\hat{j}\)

Question. Position vector of B is
(a) \(4\hat{i} + 4\hat{j}\)
(b) \(6\hat{i} + 6\hat{j}\)
(c) \(9\hat{i} + 7\hat{j}\)
(d) \(3\hat{i} + 3\hat{j}\)
Answer: (c) \(9\hat{i} + 7\hat{j}\)

Question. Find the vector \(\vec{AC}\).
(a) \(8\hat{j}\)
(b) \(-8\hat{j}\)
(c) \(8\hat{i}\)
(d) None of the options
Answer: (b) \(-8\hat{j}\)

Question. If \(\vec{A} = \hat{i} + 2\hat{j} + 3\hat{k}\), then its unit vector is
(a) \(\frac{\hat{i}}{\sqrt{14}} + \frac{2\hat{j}}{\sqrt{14}} + \frac{3\hat{k}}{\sqrt{14}}\)
(b) \(\frac{3\hat{i}}{\sqrt{14}} + \frac{2\hat{j}}{\sqrt{14}} + \frac{\hat{k}}{\sqrt{14}}\)
(c) \(\frac{2\hat{i}}{\sqrt{14}} + \frac{3\hat{j}}{\sqrt{14}} + \frac{\hat{k}}{\sqrt{14}}\)
(d) None of the options
Answer: (a) \(\frac{\hat{i}}{\sqrt{14}} + \frac{2\hat{j}}{\sqrt{14}} + \frac{3\hat{k}}{\sqrt{14}}\)

Question. If \(\vec{A} = 4\hat{i} + 3\hat{j}\) and \(\vec{B} = 3\hat{i} + 4\hat{j}\), then \(|\vec{A}| + |\vec{B}| =\)
(a) 12
(b) 13
(c) 14
(d) 10
Answer: (d) 10

Assertion & Reasoning Based MCQs

Directions: In these questions, a statement of Assertion is followed by a statement of Reason. Choose the correct answer out of the following choices:
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = 3\), \(|\vec{b}| = 4\), \(|\vec{c}| = 5\), then \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\) is equal to \(-25\).
Reason: If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), then the angle \(\theta\) between \(\vec{b}\) and \(\vec{c}\) is given by \(\cos\theta = \frac{|\vec{a}|^2 - |\vec{b}|^2 - |\vec{c}|^2}{2|\vec{b}||\vec{c}|}\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: The length of projection of the vector \(3\hat{i} - \hat{j} - 2\hat{k}\) on the vector \(\hat{i} + 2\hat{j} - 3\hat{k}\) is \(\frac{7}{\sqrt{14}}\).
Reason: The projection of a vector \(\vec{a}\) on another vector \(\vec{b}\) is \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion: \((\vec{a} \times \vec{b})^2 + (\vec{a} \cdot \vec{b})^2 \neq |\vec{a}|^2 |\vec{b}|^2\).
Reason: \(\sin^2\theta + \cos^2\theta = 1\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: If \((\vec{a} \times \vec{b})^2 + (\vec{a} \cdot \vec{b})^2 = 400\) and \(|\vec{a}| = 4\), then \(|\vec{b}| = 9\).
Reason: If \(\vec{a}\) and \(\vec{b}\) are any two vectors, then \((\vec{a} \times \vec{b})^2\) is equal to \(|\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: If \(\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}\) and \(\vec{b} = -\hat{i} + 3\hat{j} + 4\hat{k}\), then projection of \(\vec{a}\) on \(\vec{b}\) is \(\frac{3}{\sqrt{26}}\).
Reason: Projection of \(\vec{a}\) on \(\vec{b}\) is \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion: Three points with position vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) are collinear if \(\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \vec{0}\).
Reason: If \(\vec{AB} \cdot \vec{AC} = 0\), then \(\vec{AB} \perp \vec{AC}\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.